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| Description: The composition of two functions is a function. Exercise 29 of [TakeutiZaring] p. 25. |
| Ref | Expression |
|---|---|
| funco |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | moexexv 1473 |
. . . . . . 7
| |
| 2 | funmo 3607 |
. . . . . . 7
| |
| 3 | dffun6 3606 |
. . . . . . . 8
| |
| 4 | 3 | pm3.27bi 324 |
. . . . . . 7
|
| 5 | 1, 2, 4 | syl2an 456 |
. . . . . 6
|
| 6 | 5 | ancoms 438 |
. . . . 5
|
| 7 | visset 1851 |
. . . . . . 7
| |
| 8 | visset 1851 |
. . . . . . 7
| |
| 9 | 7, 8 | brco 3352 |
. . . . . 6
|
| 10 | 9 | mobii 1438 |
. . . . 5
|
| 11 | 6, 10 | sylibr 198 |
. . . 4
|
| 12 | 11 | 19.21aiv 1319 |
. . 3
|
| 13 | relco 3567 |
. . 3
| |
| 14 | 12, 13 | jctil 290 |
. 2
|
| 15 | dffun6 3606 |
. 2
| |
| 16 | 14, 15 | sylibr 198 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| This theorem is referenced by: fnco 3670 fco 3711 f1co 3742 fvco 3850 curry1 4208 curry2 4211 mapenlem1 4578 vsfval 8373 |
| This theorem was proved from axioms: ax-1 4 ax-2 5 ax-3 6 ax-mp 7 ax-7 994 ax-gen 995 ax-8 996 ax-10 998 ax-11 999 ax-12 1000 ax-13 1001 ax-14 1002 ax-17 1003 ax-4 1005 ax-5o 1007 ax-6o 1010 ax-9o 1155 ax-10o 1173 ax-16 1243 ax-11o 1251 ax-ext 1494 ax-sep 2754 ax-pow 2794 ax-pr 2832 |
| This theorem depends on definitions: df-bi 145 df-or 222 df-an 223 df-ex 1013 df-sb 1205 df-eu 1415 df-mo 1416 df-clab 1500 df-cleq 1505 df-clel 1508 df-ne 1624 df-v 1850 df-dif 2093 df-un 2094 df-in 2095 df-ss 2097 df-nul 2325 df-pw 2447 df-sn 2457 df-pr 2458 df-op 2461 df-br 2670 df-opab 2718 df-id 2889 df-xp 3239 df-rel 3240 df-cnv 3241 df-co 3242 df-fun 3247 |